Magnetic dynamics of hedgehog in icosahedral quasicrystal

Quasicrystals (QCs) possess a unique lattice structure without translational invariance, which is characterized by the rotational symmetry forbidden in periodic crystals such as the 5-fold rotation. Recent discovery of the ferromagnetic (FM) long-range order in the terbium-based QC has brought about breakthrough but the magnetic structure and dynamics remain unresolved. Here, we reveal the dynamical as well as static structure of the FM hedgehog state in the icosahedral QC. The FM hedgehog is shown to be characterized by the triple-Q state in the reciprocal-lattice \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\varvec{q}}}$$\end{document}q space. Dynamical structure factor is shown to exhibit highly structured \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\varvec{q}}}$$\end{document}q and energy dependences. We find a unique magnetic excitation mode along the 5-fold direction exhibiting the streak fine structure in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\varvec{q}}}$$\end{document}q-energy plane, which is characteristic of the hedgehog in the icosahedral QC. Non-reciprocal magnetic excitations are shown to arise from the FM hedgehog order, which emerge in the vast extent of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\varvec{q}}}$$\end{document}q-energy plane.

www.nature.com/scientificreports/ are considered to form the lattice structure of Yb in the Cd 5.7 Yb-type QC 3 . Figure 1B shows the main structure of the QC where the Tb-12 cluster, i.e., IC is located at each vertex of the icosidodecahedron with the total number of the vertices being 30. In the Cd 5.7 Yb-type QC, there exists a few other ICs as well as Tb sites located between the ICs. In this study, as a first step of analysis, we consider the Tb sites shown in Fig. 1B with the total lattice number N = 12 × 30 = 360 to get insight into the magnetic dynamics in the QC. Here, we employ the real Tb configuration for the IC (see Fig. 1A) as well as the icosidodecahedron in the 1/1 approximant crystal (AC) Au 70 Si 17 Tb 13 whose lattice structure was solved by the X-ray measurement 25 , as a typical case. The diameter of the IC is 10.56 Å. In Fig. 1B, the IC is located at 30 vertices of the τ 3 -times enlarged icosidodecahedron in the Tsai-type cluster of Au 70 Si 17 Tb 13 with τ being the golden mean τ ≡ (1 + √ 5)/2.
Minimal model in rare earth-based QC. The Tb 3+ ion with 4f 8 configuration has the ground state of the crystalline electric field (CEF) with the total angular momentum J = 6 according to the Hund's rule. The quantization axis of the CEF is the vector passing through each Tb site from the center of the IC, which is the pseudo 5-fold axis (see Fig. 1A). The detailed analysis of the CEF has revealed that the magnetic anisotropy arising from the CEF plays a key role in realizing the unique magnetic state such as the hedgehog on the IC 18,19 . Then, we consider the minimal model for the magnetism in the Tb-based QC as where J ij is the exchange interaction between the ith and jth Tb sites and S i is the "spin" operator with S i = 6 . In the second term, the unit vector ê 3 indicates the direction of the magnetic anisotropy arising from the CEF, which can be controlled by the compositions of Au and SM in Au-SM-Tb 18,19 . This model is expected to be relevant to not only the Tb-based QC but also rare-earth based QCs. In this study, to discuss the hedgehog state, ê 3 is set to be the pseudo 5-fold axis direction. For the strong limit of the magnetic anisotropy, it has been shown that the uniform long-range order of the hedgehog state is realized in the QC for Fig. S1). Each IC is characterized by the topological charge of one n TC = +1 , which is distributed quasi-periodically in Fig. 1B 18 . The hedgehog is the source of emergent field, which is regarded as monopole with the charge n TC = +1 26,27 . Magnetic excitation in QC. In the hedgehog state, "spins" are non-collinearly aligned as shown in Fig. 1A.
Hence it is convenient to introduce the local coordinate at each Tb site where the ê 3 axis is set as the ordered "spin" direction as shown in Fig. 1C (see Methods section for detail). Then, by applying the Holstein-Primakoff transformation 28 to H, the "spin" operators are transformed to the boson operators as S + i = √ 2S − n i a i , is the lowering (raising) "spin" operator and a i (a † i ) is an annihilation (creation) operator of the boson at the ith Tb site. Here the quadratic terms of the boson operators are retained because the higher order terms are considered to be irrelevant at least for the ground state.
We employ J 1 = 1.0 and J 2 = 2.3 as a typical parameter for the Tb-based QC. Actually, J 2 /J 1 = 2.3 has been experimentally identified in the model Eq. (1) for the large D limit applied to the 1/1 AC Au 72 Al 14 Tb 14 29 . We confirmed that the hedgehog state shown in Fig. 1B with the N = 360 sites under open boundary condition is realized as the ground state for D ≥ 17.85 in Eq. (1), which gives the positive excitation energy ω i for i = 1, · · · , N , as shown in Fig. 2A. The D dependence of the lowest excitation energy, i.e., the gap � ≡ ω N /(J 1 S) between the first-excited energy and the ground enegy is shown in Fig. 2B. In the spectrum, there exist several gaps, as remarkably seen in Fig. 2A as � 1 ≡ (ω 90 − ω 91 )/(J 1 S) . As D increases, the energy gap as well as 1 Static structure factor of magnetism. Then we calculate the magnetic structure factor The largest peak is located at Q 1 ≡ (1.77, 0, 1.02) Å −1 as shown in Fig. 2C. Since the alignment of the magnetic moments in the hedgehog shown in Fig. 1A is invariant under the permutation of x, y, and z axis, the same results in F s (q) as Fig. 2C are obtained by replacing (q x , q y , q z ) with (q y , q z , q x ) and also with (q z , q x , q y ) . Indeed we confirmed that the largest peak in F s (q) appears at Q 2 ≡ (0, 1.02, 1.77) and Thus the hedgehog state is characterized by the triple-Q (Q 1 , Q 2 , and Q 3 ) state. In Fig. 2C, the spots lie along the pseudo 5-fold axis indicated by the dashed line with an arrow named d * e i . Here, � d * e i (i = 1, · · · , 6) is the primitive vector of the six-dimensional reciprocal lattice space as the physical (external) space components as shown in Fig. 2C 30 . Hereafter, we express the pseudo-5 fold axis for the � d * e i direction as the d * e i line with an arrow. We note that the slope of the d * e i line for i = 2 in Fig. 2C is 1.736 reflecting the real configuration of the Tb sites in the IC 25 shown in Fig. 1A, which is known to be τ in the regular IC 30 . The slope of the d * e 3 , d * e 5 , and d * e 4 lines is the sign-reversed value of the slope of the d * e 2 , d * e 1 , and d * e 6 lines within each q z -q x , q x -q y , and q y -q z plane, respectively (see Fig. 2D).
It is noted that S xx (q) , S zz (q) , and S yy (q) have the maximum at q = Q 1 , Q 2 , and Q 3 , respectively, where S αβ (q) is defined as S αβ (q) ≡ 1 N i,j e iq·(r i −r j ) �S iα S jβ � (α = x, y, and z).
Dynamical structure factor of magnetism. The dynamical magnetic structure factor is defined as Here, |GS� is the ground state satisfying α i |GS� = 0 and E 0 is the ground-state energy. We set η = 10 −6 .
(2) www.nature.com/scientificreports/ The result of S xx (q, ω) for q along the d * e 2 line in the q z -q x plane is shown in Fig. 3A. The spectra appear at ω/(J 1 S) = 0 (see Fig. 3B) with strong intensity of ∼ O(10 8 ) and also appear above the energy gap with intensity of ∼ O(10 4 ) . The energy gap in the excitation spectra reflects the magnetic anisotropy arising from the CEF. For ω/(J 1 S) > � , the large intensity appears at the energy ω 90 /(J 1 S) = 22.90 , where the highest peak appears at the Ŵ point. At ω = 0 , i.e., elastic energy, the maximum peak appears at q = Q 1 , as shown in Fig. 3B indicated by the dashed line. In the ω dependence of S xx (q, ω) , spiky peak structures appear as shown in Fig. 3C for q = Q 1 . These results indicate that the peak S xx (Q 1 ) is governed by the elastic contribution S xx (Q 1 , 0) , which is understandable from the sum rule with respect to ω as S xx (Q 1 ) = 1 2π dωS xx (Q 1 , ω). This is in sharp contrast to the result recently reported in the uniform long-range order of the ferrimagnetic state in the icosahedral QC 31 . Namely, the high-intensity peak appears at the ordered vector q = 0 and the lowest CEF excitation energy ω/(|J 1 |S) = � , from which the high-intensity peaks are continuously formed in the dynamical structure factor, giving rise to the pseudo-magnon mode 31 .
Then we search the q dependence of S xx (q, ω) for ω = ω 90 where the large intensities appear as shown in Fig. 3A. Consequently, we identify that the maximum is located at q 0 ≡ (2.169, 3.436, 1.442) Å −1 . Around q = q 0 , we find that a series of the packet structures appears along the pseudo 5-fold axis direction, as shown in Fig. 4A where the d * e 2 line and d * e 3 line through q = q 0 is illustrated by the dashed line in the q z -q x plane with q y = 3.436 Å −1 . The peak in the central packet gives the maximum S xx (q 0 , ω 90 ) = 1.027 × 10 5 . A series of packet structures with sub-leading intensity is also aligned along the pseudo 5-fold axis direction. For slightly larger ω than ω 90 , the packets still appear along the d * e 2 line at slightly different positions as shown in Fig. 4B, which suggests the magnetic excitation propagating along the pseudo 5-fold direction. Figure 4C shows S xx (q, ω) for q along the d * e 2 line through q = q 0 . A series of the packet structures remarkably appears at the lower edge ω 90 with strong intensity, which continuously forms the streak with fine structure down to the lower-ω region as also seen in the intensity plot in Fig. 4D.
Interestingly, we find that a series of the packet structures is the reflection of the bottom of the continuous mode periodic along the d * e 2 line in the q-ω plane as shown in Fig. 4D. The period of the streak structure is evaluated as q ∼ 0.6 Å −1 in the reciprocal space. From the relation of the wavenumber and wavelength �q = 2π/ , the scale of the wavelength is estimated to be ∼ 10 Å . It turns out that this corresponds to the diameter of the IC d = 10.56 Å (see Fig. 1A). Since the hedgehog is the magnetic texture on the IC, the excitation gives rise to the dynamics whose intensity decreases with periodicity �q ∼ 2π/d with distance from q 0 in the reciprocal space of the QC. A series of the packet structure as well as the intensity streak in the q-ω plane also appears along the d * e 3 direction (Supplementary information, Fig. S3). The emergence of the intensity streak with fine structure in the q-ω plane indicates unique excitation mode along the 5-fold axis direction, which is considered to be characteristic of the hedgehog in the icosahedral QC.

Non-reciprocal magnetic excitation in QC.
To further clarify the general property of the dynamics of the hedgehog state, we show S xx (q, ω) at ω/(J 1 S) = 23.18 for q along the pink lines in the cube whose side is parallel to the 2-fold axis q α ∈ [0, 2.56] Å −1 (α = x, y, and z) in the inset of Fig. 5A. Here we also plot S xx (−q, ω) www.nature.com/scientificreports/ along the green line in the inset of Fig. 5A. We see remarkable differences in the intensity for q and −q . In Fig. 5B, we plot |S xx (q, ω) − S xx (−q, ω)| for q along the pink lines in the inset of Fig. 5A. The finite values indicate that S xx (q, ω) = S xx (−q, ω) . These results indicate that non-reciprocal magnetic excitation appears in the hedgehog state in the QC. This is, to our best knowledge, the first discovery of the non-reciprocal magnetic excitation in the topological magnetic long-range order in QC. We confirmed that non-reciprocal magnetic excitation does not appear in the case of the collinear magnetic order in the QC (Supplementary information, Fig. S4). This implies that the noncollinear and noncoplanar  www.nature.com/scientificreports/ magnetic structure on the IC of the hedgehog (Fig. 1A) is the origin of the nonreciprocal excitation. Recently, non-reciprocal magnetic excitation from the uniform ferrimagnetic order (characterized by the zero topological charge n TC = 0 ) in the icosahedral QC has been shown to appear 31 . These results suggest that non-reciprocal excitation is common character of the noncollinear and noncoplanar alignment of the magnetic moments on the IC. As shown in Fig. 5B, emergence of many spiky peaks with fine structure as continuum are the consequence of the QC structure, which is in sharp contrast to the magnon branch in periodic crystals as the collective mode. This gives rise to the emergence of nonreciprocity as continuum in the vast extent of the q-ω plane (see Fig. 5B), whose feature is unique to the QC.

Summary and discussion.
We have revealed the dynamical as well as static property of the hedgehog state in the QC. The FM hedgehog state is shown to be characterized by the triple-Q state. The magnetic dynamical structure factor shows highly structured energy and momentum dependences unique to the QC. We have discovered the magnetic excitation mode along the pseudo 5-fold axis direction. A series of the packet structure in the dynamical structure factor is found to exist, which is shown to be the reflection of the periodic streak structure in the reciprocal lattice q-energy ω plane. Non-collinear and non-coplanar magnetic alignment of the hedgehog state gives rise to non-reciprocal magnetic excitations which appear in the vast extent of the energy and momentum plane.
In the uniform long-range order of the ferrimagnetic state, the high-intensity peaks appear continuously from the ordered vector q = 0 and the lowest CEF excitation energy ω = �|J 1 |S , which are identified as the pseudo-magnon mode 31 . On the contrary, in the dynamical structure factor for the uniform hedgehog order, the high-intensity peaks do not appear at the ordered vector Q i for i = 1 , 2, and 3 beyond the CEF excitation energy. This implies that the peak in the static structure factor at the triple Q vector q = Q i is governed by the elastic (ω = 0) contribution of the dynamical structure factor for the uniform hedgehog order.
The streak structure with periodicity characterized by the wavelength corresponding to the diameter of the IC in the q-ω plane is considered to be the unique character of the excitation from the uniform hedgehog order. To establish this point, the systematic analysis of the dynamical structure factor in the magnetically ordered states in the icosahedral QC is necessary, which is left for future studies.
The non-reciprocal magnetic excitation has also been found to emerge in the uniform ferrimagnetic order 31 . Hence, as noted above, non-reciprocity is considered to be general feature of the excitation from the non-collinear and non-coplanar magnetic texture on the IC.
Our results are useful not only for resolving the magnetic structure of the long-range order discovered recently in Tb-based icosahedral QC, but also for future neutron measurements of the magnetic dynamics in the QC. So far, the dynamical structure factor in the magnetically ordered phase in the QC has not been reported. It is expected that present study stimulates future experiments to detect the dynamical property in the QC and also in the approximant crystal.

Theory of Magnetic Excitation in QC.
Magnetic excitation from the uniform hedgehog long-range order in the QC can be calculated by transforming spin operators in the model (1) into boson operators. Since the hedgehog is a noncoplanar magnetic state, it is convenient to introduce the local coordinate at each Tb site 32 . The unit vectors in the global xyz coordinate r 1 =x , r 2 =ŷ , and r 3 =ẑ are expressed by the local orthogonal coordinate with the unit vector ê i 3 , whose direction is indicated by the polar angles (θ i , φ i ) , as (see Fig. 1C). Here, R i is the rotation matrix defined as Then, the first term in Eq. (1) is expressed as where S + i and S − i are raising and lowering "spin" operators, respectively, we apply the Holstein-Primakoff transformation 28 to H. Namely, "spin" operators are expressed by the boson operators as S + i = √ 2S − n i a i , S − i = a † i √ 2S − n i and S i ·ê i 3 = S − n i with n i ≡ a † i a i . We retain the quadratic terms with respect to a † i and a i , which are considered to be at least valid for the ground state. In the noncollinear magnetic state as the hedgehog, anomalous terms such as a † i a † j and a i a j appear. The resultant H is expressed as where χ † = (a † 1 , a † 2 , · · · , a † N ) and is the 2N × 2N matrix. By performing the para unitary transformation (4) r α = R i αβê i β (5) R i = cos θ i cos φ i − sin φ i sin θ i cos φ i cos θ i sin φ i cos φ i sin θ i sin φ i − sin θ i 0 cos θ i .

Data availability
All the data supporting the findings are available from the corresponding author upon reasonable request.